A Clustering-Based Variable Ordering Framework for Relaxed Decision Diagrams for Maximum Weighted Independent Set Problem

Abstract

Efficient exact algorithms for Discrete Optimization (DO) rely heavily on strong primal and dual bounds. Relaxed Decision Diagrams (DDs) provide a versatile mechanism for deriving such dual bounds by compactly over-approximating the solution space through node merging. However, the quality of these relaxed diagrams, i.e. the tightness of the resulting dual bounds, depends critically on the variable ordering and the merging decisions executed during compilation. While dynamic variable ordering heuristics effectively tighten bounds, they often incur computational overhead when evaluated globally across the entire variable set. To mitigate this trade-off, this work introduces a novel clustering-based framework for variable ordering. Instead of applying dynamic ordering heuristics to the full set of unfixed variables, we first partition variables into clusters. We then leverage this structural decomposition to guide the ordering process, significantly reducing the heuristic's search space. Within this framework, we investigate two distinct strategies: Cluster-to-Cluster, which processes clusters sequentially using problem-specific aggregate criteria (such as cumulative vertex weights in the Maximum Weighted Independent Set Problem (MWISP)), and Pick-and-Sort, which iteratively selects and sorts representative variables from each cluster to balance local diversity with heuristic guidance. Later on, developing some theoretical results on the growth of the size of DDs for MWISP we propose two different policies for setting the number of clusters within the proposed framework. We embed these strategies into a DD-based branch-and-bound algorithm and evaluate them on the MWISP. Across benchmark instances, the proposed methodology consistently reduces computational costs compared to standard dynamic variable ordering baseline.